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I am trying to better understand how the "Reduction of Derivatives" (see e.g. Callen Chpt. 7) works in thermodynamics for multi-component systems. All questions are at the bottom.

In a closed system, or a single component open system, there appears to me to be a standard way to reduce thermodynamic derivatives into measurable properties that avoids introducing any derivatives in terms of the chemical potential. However, this approach seems to breakdown in the multi-component case. I've tried to lay out the logic of what I mean below:

One Component Case

Fundamental Equation:

$$dU=TdS-PdV+\mu dm$$

Gibbs-Duhem Equation: $$d\mu=-\frac{S}{m}dT+\frac{V}{m}dP$$

To me, this implies that any derivative where $\mu$ shows up can be reduced to derivatives of $T$, and $P$ in the same way that any derivative where $U$ shows up could be reduced to derivatives of $S, V$ and $m$ by the Fundamental Equation.

For example, in the Gibbs Free Energy Representation $(T,P,m)$ one could reduce the following partial derivative into an equation where all partial derivatives contain only $P,T$, and $V$ (or equivalently $\alpha, \kappa_T, c_p$): $$\frac{\partial{S}}{\partial{m}}\bigg\rvert_{T,V}=\frac{S}{m}-\frac{V\alpha}{m\kappa_T}$$

My understanding is that this equation, and any other arbitrary derivative for this system, will never contain a derivative in terms of $\mu$ because of the particular form of the Gibbs-Duhem equation.

Two Component Case

Fundamental Equation for two components: $$dU=TdS-PdV+\mu_1 dm_1+\mu_2 dm_2$$

The Gibbs-Duhem Equation now takes on the form: $$d\mu_1=-\frac{S}{m_1}dT+\frac{V}{m_1}dP-\frac{m_2}{m_1}d\mu_2$$

Questions

Am I correct in thinking that because of the new form of the Gibbs-Duhem Equation, the reduction of any arbitrary derivative in the Gibbs Free energy representation $(T,P,m_1,m_2)$ will inevitably end up with a derivative of $\mu_2$ (in addition to the standard $P,T$, and $V$)?

Is there any standard representation that will reduce an arbitrary derivative into a form that only contains derivatives in terms of things I can measure directly (e.g. $T,P$, and $V$)?

Doesn't this problem generalize to the n-component case and the n-phase case? Does anyone have any perspective on what is typically done in these situations, or is it standard to just start including models for the chemical potential?

Any help would be appreciated!

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